Reciprocal Trig Identities

The reciprocal identities are derived from the definitions of the 3 basic trig functions. They are based on the definition of three of the functions as reciprocals of the basic three - sine, cosine, and tangent.

For any angle θ:

sin θ =

1

csc θ

csc θ =

1

sin θ

cos θ =

1

sec θ

sec θ =

1

cos θ

tan θ =

1

cot θ

cot θ =

1

tan θ

Quotient Trig Identities

The quotient identities are derived from the definitions of the 3 basic trig functions in a unit circle and substitution from the definitions.

In a unit circle, sin = y, cos = x, and tan = y/x. Using substitution, for any angle θ:

tan θ =

sin θ

cos θ

cot θ =

cos θ

sin θ

Pythagorean Trig Identities

The pythagorean identities are based on the Pythagorean Theorem, which states that in a right triangle with legs a and b and hypotenuse c:

a2 + b2 = c2

Using substitution into the Phythagorean Theorem from the trig definitions, for any angle θ:

sin2 + cos2 = 1

From this identity, also called the Fundamental Theorem of Trig, can be derived the other two related identities: